For any \(i\in I\) , Lusztig ([1]}) introduced derivations \({_ir}:\mathbf {U}^-\rightarrow \mathbf {U}^-\)  and \({r_i}:\mathbf {U}^-\rightarrow \mathbf {U}^-\)  such that
\({_ir}^{m}(xy)=\sum _{t=0}^{m}v^{(\nu -ti,(m-t)i)+t(m-t)}\frac{[m]_{v}!}{[t]_{v}![m-t]_{v}!}{_ir}^{t}(x){_ir}^{m-t}(y)\) 
